MathDB

4

Part of 2003 IMO Shortlist

Problems(3)

IMO ShortList 2003, geometry problem 4

Source: IMO ShortList 2003, geometry problem 4

10/4/2004
Let Γ1\Gamma_1, Γ2\Gamma_2, Γ3\Gamma_3, Γ4\Gamma_4 be distinct circles such that Γ1\Gamma_1, Γ3\Gamma_3 are externally tangent at PP, and Γ2\Gamma_2, Γ4\Gamma_4 are externally tangent at the same point PP. Suppose that Γ1\Gamma_1 and Γ2\Gamma_2; Γ2\Gamma_2 and Γ3\Gamma_3; Γ3\Gamma_3 and Γ4\Gamma_4; Γ4\Gamma_4 and Γ1\Gamma_1 meet at AA, BB, CC, DD, respectively, and that all these points are different from PP. Prove that AB⋅BCAD⋅DC=PB2PD2. \frac{AB\cdot BC}{AD\cdot DC}=\frac{PB^2}{PD^2}.
geometryparallelogramhomothetyInversionIMO Shortlistgeometry solvedlengths
When 11...1 22...2 5 is a perfect square

Source: German TST 2004, IMO ShortList 2003, number theory problem 4

5/19/2004
Let b b be an integer greater than 5 5. For each positive integer n n, consider the number x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, written in base b b.
Prove that the following condition holds if and only if b \equal{} 10: there exists a positive integer M M such that for any integer n n greater than M M, the number xn x_n is a perfect square.
Proposed by Laurentiu Panaitopol, Romania
modular arithmeticnumber theorydecimal representationPerfect SquareIMO Shortlist
IMO ShortList 2003, combinatorics problem 4

Source: Problem 5 of the German pre-TST 2004, written in December 03

5/17/2004
Let x1,…,xnx_1,\ldots, x_n and y1,…,yny_1,\ldots, y_n be real numbers. Let A=(aij)1≤i,j≤nA = (a_{ij})_{1\leq i,j\leq n} be the matrix with entries aij={1,if xi+yj≥0;0,if xi+yj<0.a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases} Suppose that BB is an n×nn\times n matrix with entries 00, 11 such that the sum of the elements in each row and each column of BB is equal to the corresponding sum for the matrix AA. Prove that A=BA=B.
linear algebramatrixcombinatoricsIMO Shortlist